Загрузил Pavel Tsygankov

Turbulence Notes Fluent

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Modeling Turbulent
Flows
Introductory FLUENT Training
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What is Turbulence?
‹
Unsteady, irregular (aperiodic) motion in which transported quantities
(mass, momentum, scalar species) fluctuate in time and space
z
z
‹
Fluid properties and velocity exhibit random variations
z
z
‹
Identifiable swirling patterns characterize turbulent eddies.
Enhanced mixing (matter, momentum, energy, etc.) results
Statistical averaging results in accountable, turbulence related transport
mechanisms.
This characteristic allows for turbulence modeling.
Contains a wide range of turbulent eddy sizes (scales spectrum).
z
The size/velocity of large eddies is on the order of mean flow.
„
z
Large eddies derive energy from the mean flow
Energy is transferred from larger eddies to smaller eddies
„
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In the smallest eddies, turbulent energy is converted to internal energy by
viscous dissipation.
6-2
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Is the Flow Turbulent?
External Flows
ρU L
µ
L = x, d , d h , etc.
where Re L =
Re x ≥ 500,000 along a surface
Re d ≥ 20,000 around an obstacle
Other factors such as free-stream
turbulence, surface conditions, and
disturbances may cause transition
to turbulence at lower Reynolds
numbers
Internal Flows
Re d h ≥ 2,300
Natural Convection
2
3
3
ρ
C
β
g
L
∆T
β
g
L
∆
T
Ra
p
9
≥ 10
=
where Ra =
is the Rayleigh number
να
µk
Pr
ν µCp
Pr = =
is the Prandtl number
k
α
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Turbulent Flow Structures
Small
structures
Large
structures
Energy Cascade
Richardson (1922)
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Overview of Computational Approaches
‹
Reynolds-Averaged Navier-Stokes (RANS) models
z
z
z
‹
Large Eddy Simulation (LES)
z
z
‹
Solves the spatially averaged N-S equations. Large eddies are directly resolved, but
eddies smaller than the mesh are modeled.
Less expensive than DNS, but the amount of computational resources and efforts
are still too large for most practical applications.
Direct Numerical Simulation (DNS)
z
z
z
‹
Solve ensemble-averaged (or time-averaged) Navier-Stokes equations
All turbulent length scales are modeled in RANS.
The most widely used approach for calculating industrial flows.
Theoretically, all turbulent flows can be simulated by numerically solving the full
Navier-Stokes equations.
Resolves the whole spectrum of scales. No modeling is required.
But the cost is too prohibitive! Not practical for industrial flows - DNS is not
available in Fluent.
There is not yet a single, practical turbulence model that can reliably predict
all turbulent flows with sufficient accuracy.
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Turbulence Models Available in FLUENT
RANS based
models
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One-Equation Models
Spalart-Allmaras
Two-Equation Models
Standard k–ε
RNG k–ε
Realizable k–ε
Standard k–ω
SST k–ω
Reynolds Stress Model
Detached Eddy Simulation
Large Eddy Simulation
6-6
Increase in
Computational
Cost
Per Iteration
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RANS Modeling – Time Averaging
‹
Ensemble (time) averaging may be used to extract the mean flow properties
from the instantaneous ones:
1
ui (x, t ) = lim
N →∞ N
N
∑
ui
(n )
(x, t )
ui′(x, t )
n =1
ui (x, t )
ui (x, t ) = ui (x, t ) + ui′(x, t )
Instantaneous
component
‹
ui (x, t )
Example: Fully-Developed
Turbulent Pipe Flow
Velocity Profile
Time-average Fluctuating
component
component
The Reynolds-averaged momentum equations are as follows
 ∂u
∂u 
∂
∂p
ρ i + uk i  = −
+
∂xk 
∂xi ∂x j
 ∂t
z
 ∂ui
µ
 ∂x
j

 ∂ Rij
+
 ∂x
j

Rij = −ρui′u ′j
(Reynolds stress tensor)
The Reynolds stresses are additional unknowns introduced by the averaging
procedure, hence they must be modeled (related to the averaged flow quantities) in
order to close the system of governing equations.
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The Closure Problem
‹
The RANS models can be closed in one of the following ways
(1) Eddy Viscosity Models (via the Boussinesq hypothesis)
 ∂ui ∂u j  2 ∂uk
2
 − µT
+
Rij = −ρ ui′u′j = µT 
δij − ρ k δij
 ∂x ∂x  3 ∂x
3
i 
k
 j
z
Boussinesq hypothesis – Reynolds stresses are modeled using an eddy (or
turbulent) viscosity, µT. The hypothesis is reasonable for simple turbulent shear
flows: boundary layers, round jets, mixing layers, channel flows, etc.
(2) Reynolds-Stress Models (via transport equations for Reynolds stresses)
z
z
Modeling is still required for many terms in the transport equations.
RSM is more advantageous in complex 3D turbulent flows with large streamline
curvature and swirl, but the model is more complex, computationally intensive,
more difficult to converge than eddy viscosity models.
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Calculating Turbulent Viscosity
‹
Based on dimensional analysis, µT can be determined from a
turbulence time scale (or velocity scale) and a length scale.
z
z
z
‹
k = ui′ui′ 2
Turbulent kinetic energy [L2/T2]
Turbulence dissipation rate [L2/T3] ε = ν ∂ui′ ∂x j (∂ui′ ∂x j + ∂u′j ∂xi )
Specific dissipation rate [1/T]
ω=ε k
Each turbulence model calculates µT differently.
z
Spalart-Allmaras:
„
z
Standard k–ε, RNG k–ε, Realizable k–ε
„
z
Solves a transport equation for a modified
turbulent viscosity.
Solves transport equations for k and ε.
Standard k–ω, SST k–ω
„
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Solves transport equations for k and ω.
6-9
µT = f (~
ν)
 ρk2 

µT = f 
 ε 
 ρk 
µT = f  
 ω
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The Spalart-Allmaras Model
‹
Spalart-Allmaras is a low-cost RANS model solving a transport equation for a
modified eddy viscosity.
z
‹
‹
‹
Mainly intended for aerodynamic/turbomachinery applications with mild
separation, such as supersonic/transonic flows over airfoils, boundary-layer
flows, etc.
Embodies a relatively new class of one-equation models where it is not
necessary to calculate a length scale related to the local shear layer thickness.
Designed specifically for aerospace applications involving wall-bounded
flows.
z
z
‹
When in modified form, the eddy viscosity is easy to resolve near the wall.
Has been shown to give good results for boundary layers subjected to adverse
pressure gradients.
Gaining popularity for turbomachinery applications.
This model is still relatively new.
z
z
No claim is made regarding its applicability to all types of complex engineering
flows.
Cannot be relied upon to predict the decay of homogeneous, isotropic turbulence.
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The k–ε Turbulence Models
‹
Standard k–ε (SKE) model
z
z
z
z
The most widely-used engineering turbulence model for industrial applications
Robust and reasonably accurate
Contains submodels for compressibility, buoyancy, combustion, etc.
Limitations
„
„
‹
The ε equation contains a term which cannot be calculated at the wall. Therefore,
wall functions must be used.
Generally performs poorly for flows with strong separation, large streamline
curvature, and large pressure gradient.
Renormalization group (RNG) k–ε model
z
z
Constants in the k–ε equations are derived using renormalization group theory.
Contains the following submodels
„
„
„
z
Differential viscosity model to account for low Re effects
Analytically derived algebraic formula for turbulent Prandtl / Schmidt number
Swirl modification
Performs better than SKE for more complex shear flows, and flows with high strain
rates, swirl, and separation.
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The k–ε Turbulence Models
‹
Realizable k–ε (RKE) model
z
The term realizable means that the model satisfies certain mathematical
constraints on the Reynolds stresses, consistent with the physics of
turbulent flows.
„ Positivity of normal stresses: ui′u ′j > 0
„
z
z
( )
Schwarz’ inequality for Reynolds shear stresses: ui′u′j
2
≤ ui2u 2j
Neither the standard k–ε model nor the RNG k–ε model is realizable.
Benefits:
„
„
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More accurately predicts the spreading rate of both planar and round jets.
Also likely to provide superior performance for flows involving rotation,
boundary layers under strong adverse pressure gradients, separation, and
recirculation.
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The k–ω Turbulence Models
‹
The k–ω family of turbulence models have gained popularity mainly because:
z
z
‹
The model equations do not contain terms which are undefined at the wall, i.e. they
can be integrated to the wall without using wall functions.
They are accurate and robust for a wide range of boundary layer flows with
pressure gradient.
FLUENT offers two varieties of k–ω models.
z
Standard k–ω (SKW) model
„
„
z
Most widely adopted in the aerospace and turbo-machinery communities.
Several sub-models/options of k–ω: compressibility effects, transitional flows and
shear-flow corrections.
Shear Stress Transport k–ω (SSTKW) model (Menter, 1994)
„
„
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The SST k–ω model uses a blending function to gradually transition from the
standard k–ω model near the wall to a high Reynolds number version of the k–ε
model in the outer portion of the boundary layer.
Contains a modified turbulent viscosity formulation to account for the transport
effects of the principal turbulent shear stress.
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Large Eddy Simulationn
‹
Large Eddy Simulation (LES)
z
LES has been most successful for high-end applications where the RANS models
fail to meet the needs. For example:
„
„
„
‹
Implementations in FLUENT:
z
Subgrid scale (SGS) turbulent models:
„
„
„
„
z
‹
‹
‹
Combustion
Mixing
External Aerodynamics (flows around bluff bodies)
Smagorinsky-Lilly model
Wall-Adapting Local Eddy-Viscosity (WALE)
Dynamic Smagorinsky-Lilly model
Dynamic Kinetic Energy Transport
Detached eddy simulation (DES) model
LES is applicable to all combustion models in FLUENT
Basic statistical tools are available: Time averaged and RMS values of solution
variables, built-in fast Fourier transform (FFT).
Before running LES, consult guidelines in the “Best Practices For LES”
(containing advice for meshing, subgrid model, numerics, BCs, and more)
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Law of the Wall and Near-Wall Treatments
‹
Dimensionless velocity data from a wide variety of turbulent duct and
boundary-layer flows are shown here:
τw
Wall shear
Uτ =
stress
ρ
u
yUτ
+
+
u
=
y =
Uτ
ν
where y is the normal
distance from the wall
‹
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6-15
For equilibrium turbulent
boundary layers, walladjacent cells in the log-law
region have known velocity
and wall shear stress data
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Wall Boundary Conditions
‹
‹
The k–ε family and RSM models are not valid in the near-wall region,
whereas Spalart-Allmaras and k–ω models are valid all the way to the
wall (provided the mesh is sufficiently fine). To work around this, we
can take one of two approaches.
Wall Function Approach
z
z
z
z
‹
Standard wall function method is to take advantage of the fact that (for
equilibrium turbulent boundary layers), a log-law correlation can supply
the required wall boundary conditions (as illustrated in the previous slide).
Non-equilibrium wall function method attempts to improve the results for
flows with higher pressure gradients, separations, reattachment and
stagnation.
Similar laws are also constructed for the energy and species equations.
Benefit: Wall functions allow the use of a relatively coarse mesh in the
near-wall region.
Enhanced Wall Treatment Option
z
z
z
z
Combines a blended law-of-the wall and a two-layer zonal model.
Suitable for low-Re flows or flows with complex near-wall phenomena
Turbulence models are modified for the inner layer.
Generally requires a fine near-wall mesh capable of resolving the viscous
sublayer (at least 10 cells within the “inner layer”)
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outer layer
inner layer
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Placement of The First Grid Point
‹
‹
For standard or non-equilibrium wall functions, each wall-adjacent cell
+
centroid should be located within the log-law layer y p ≈ 30 − 300
For enhanced wall treatment (EWT), each wall-adjacent cell centroid should
+
be located within the viscous sublayer y p ≈ 1
z
‹
EWT can automatically accommodate cells placed in the log-law layer
How to estimate the size of wall-adjacent cells before creating the grid:
y +p =
z
y p uτ
ν
yp =
Uτ =
uτ
τw
Cf
= Ue
2
ρ
The skin friction coefficient can be estimated from empirical correlations:
Flat plate:
‹
⇒
y +p ν
Cf
0.037
≈
2
Re1L 5
Duct:
Cf
2
≈
0.039
Re1D4h
Use postprocessing tools (XY plot or contour plot) to to double check the nearwall grid placement after the flow pattern has been established.
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Near-Wall Modeling: Recommended Strategy
‹
Use standard or non-equilibrium wall functions for most high
Reynolds number applications (Re > 106) for which you cannot afford
to resolve the viscous sublayer.
z
z
‹
You may consider using enhanced wall treatment if:
z
z
‹
There is little gain from resolving the viscous sublayer. The choice of
core turbulence model is more important.
Use non-equilibrium wall functions for mildly separating, reattaching, or
impinging flows.
The characteristic Reynolds number is low or if near wall characteristics
need to be resolved.
The physics and near-wall mesh of the case is such that y+ is likely to vary
significantly over a wide portion of the wall region.
Try to make the mesh either coarse or fine enough to avoid placing the
wall-adjacent cells in the buffer layer (5 < y+ < 30).
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Inlet and Outlet Boundary Conditions
‹
‹
When turbulent flow enters a domain at inlets or outlets (backflow), boundary conditions for k,
ε, ω and/or ui′u ′j must be specified, depending on which turbulence model has been selected
Four methods for directly or indirectly specifying turbulence parameters:
z Explicitly input k, ε, ω, or
„
„
z
This is the only method that allows for profile definition.
See user’s guide for the correct scaling relationships among them.
Turbulence intensity and length scale
„
Length scale is related to size of large eddies that contain most of energy.
Œ For boundary layer flows: l ≈ 0.4 δ99
Œ For flows downstream of grid: l ≈ opening size
z
Turbulence intensity and hydraulic diameter
„
z
Turbulence intensity and turbulent viscosity ratio
„
‹
‹
Ideally suited for internal (duct and pipe) flows
For external flows: 1 < µt/µ < 10
Turbulence intensity depends on upstream conditions: I =
u′ 1
≈
U U
2k
< 20 %
3
Stochastic inlet boundary conditions for LES and RANS can be generated by using spectral
synthesizer or vortex method
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GUI for Turbulence Models
Define
Models
Viscous…
Inviscid, Laminar,
or Turbulent
Define
Boundary Conditions…
Turbulence
Model Options
Near Wall
Treatments
Additional
Options
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Example #1 – Turbulent Flow Past a Blunt Plate
Reynolds-Stress
model (“exact”)
Standard k-ε
model
The Standard k–ε model is
known to give spuriously
large TKE on the font face
of the plate
Contour plots
of turbulent
kinetic energy
(TKE)
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Example #1 – Turbulent Flow Past a Blunt Plate
Skin Friction Coefficient
Predicted separation bubble:
Standard k-ε (SKE)
Realizable k-ε (RKE)
SKE severely underpredicts the size of
the separation bubble, while RKE model
predicts the size exactly.
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Experimentally observed reattachment
point is at x/d = 4.7
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Example #2 – Turbulent Flow in a Cyclone
0.1 m
‹
‹
‹
‹
40,000-cell hexahedral mesh
High-order upwind scheme
was used.
Computed using SKE, RNG,
RKE and RSM (second
moment closure) models with
the standard wall functions
Represents highly swirling
flows (Wmax = 1.8 Uin)
0.12 m
Uin = 20 m/s
0.97 m
0.2 m
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Example #2 – Turbulent Flow in a Cyclone
‹
Tangential velocity profile predictions at 0.41 m below the vortex
finder
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Example #3 – Flow Past a Square Cylinder (LES)
Drag
Coefficient
Strouhal
Number
Dynamic Smagorinsky
2.28
0.130
Dynamic TKE
2.22
0.134
2.1 – 2.2
0.130
Exp.(Lyn et al., 1992)
Time-averaged streamwise
velocity along the wake centerline
Iso-Contours of Instantaneous
Vorticity Magnitude
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(ReH = 22,000)
CL spectrum
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Example #3 – Flow Past a Square Cylinder (LES)
Streamwise mean velocity
along the wake centerline
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Streamwise normal stress
along the wake centerline
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Summary: Turbulence Modeling Guidelines
‹
Successful turbulence modeling requires engineering judgment of:
z
z
z
Flow physics
Computer resources available
Project requirements
„
„
z
‹
Accuracy
Turnaround time
Near-wall treatments
Modeling procedure
z
z
z
z
z
Calculate characteristic Re and determine whether the flow is turbulent.
Estimate wall-adjacent cell centroid y+ before generating the mesh.
Begin with SKE (standard k-ε) and change to RKE, RNG, SKW, SST or
V2F if needed. Check the tables in the appendix as a starting guide.
Use RSM for highly swirling, 3-D, rotating flows.
Use wall functions for wall boundary conditions except for the low-Re
flows and/or flows with complex near-wall physics.
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Appendix
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RANS Turbulence Model Descriptions
Model
Description
Spalart –
Allmaras
A single transport equation model solving directly for a modified turbulent viscosity. Designed specifically
for aerospace applications involving wall-bounded flows on a fine near-wall mesh. FLUENT’s
implementation allows the use of coarser meshes. Option to include strain rate in k production term
improves predictions of vortical flows.
Standard k–ε
The baseline two-transport-equation model solving for k and ε. This is the default k–ε model. Coefficients
are empirically derived; valid for fully turbulent flows only. Options to account for viscous heating,
buoyancy, and compressibility are shared with other k–ε models.
RNG k–ε
A variant of the standard k–ε model. Equations and coefficients are analytically derived. Significant
changes in the ε equation improves the ability to model highly strained flows. Additional options aid in
predicting swirling and low Reynolds number flows.
Realizable k–ε
A variant of the standard k–ε model. Its “realizability” stems from changes that allow certain mathematical
constraints to be obeyed which ultimately improves the performance of this model.
Standard k–ω
A two-transport-equation model solving for k and ω, the specific dissipation rate (ε / k) based on Wilcox
(1998). This is the default k–ω model. Demonstrates superior performance for wall-bounded and low
Reynolds number flows. Shows potential for predicting transition. Options account for transitional, free
shear, and compressible flows.
SST k–ω
A variant of the standard k–ω model. Combines the original Wilcox model for use near walls and the
standard k–ε model away from walls using a blending function. Also limits turbulent viscosity to guarantee
that τT ~ k. The transition and shearing options are borrowed from standard k–ω. No option to include
compressibility.
Reynolds Stress
Reynolds stresses are solved directly using transport equations, avoiding isotropic viscosity assumption of
other models. Use for highly swirling flows. Quadratic pressure-strain option improves performance for
many basic shear flows.
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RANS Turbulence Model Behavior and Usage
Model
Behavior and Usage
Spalart-Allmaras
Economical for large meshes. Performs poorly for 3D flows, free shear flows, flows with strong
separation. Suitable for mildly complex (quasi-2D) external/internal flows and boundary layer flows
under pressure gradient (e.g. airfoils, wings, airplane fuselages, missiles, ship hulls).
Standard k–ε
Robust. Widely used despite the known limitations of the model. Performs poorly for complex flows
involving severe pressure gradient, separation, strong streamline curvature. Suitable for initial
iterations, initial screening of alternative designs, and parametric studies.
RNG k–ε
Suitable for complex shear flows involving rapid strain, moderate swirl, vortices, and locally transitional
flows (e.g. boundary layer separation, massive separation, and vortex shedding behind bluff bodies, stall
in wide-angle diffusers, room ventilation).
Realizable k–ε
Offers largely the same benefits and has similar applications as RNG. Possibly more accurate and easier
to converge than RNG.
Standard k–ω
Superior performance for wall-bounded boundary layer, free shear, and low Reynolds number flows.
Suitable for complex boundary layer flows under adverse pressure gradient and separation (external
aerodynamics and turbomachinery). Can be used for transitional flows (though tends to predict early
transition). Separation is typically predicted to be excessive and early.
SST k–ω
Offers similar benefits as standard k–ω. Dependency on wall distance makes this less suitable for free
shear flows.
Reynolds Stress
Physically the most sound RANS model. Avoids isotropic eddy viscosity assumption. More CPU time
and memory required. Tougher to converge due to close coupling of equations. Suitable for complex
3D flows with strong streamline curvature, strong swirl/rotation (e.g. curved duct, rotating flow
passages, swirl combustors with very large inlet swirl, cyclones).
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The Spalart-Allmaras Turbulence Model
‹
‹
A low-cost RANS model solving an equation for the modified eddy
viscosity, ~ν
2


~
~
~




Dν
∂
ν
∂ν

 ∂
~


(
)
C
= Gν 
µ
+
ρ
ν
+
ρ


 − Yν + S ~ν
b2 

Dt
∂x j 
 ∂x j 
 ∂x j  
Eddy viscosity is obtained from
µ = ρ~
νf
t
‹
‹
3
~
(
ν / ν)
f v1 = ~ 3
(ν / ν ) + Cv31
v1
The variation of ~
ν very near the wall is easier to resolve than k and ε.
Mainly intended for aerodynamic/turbomachinery applications with
mild separation, such as supersonic/transonic flows over airfoils,
boundary-layer flows, etc.
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RANS Models - Standard k–ε (SKE) Model
‹
Transport equations for k and ε:
D
∂ 
(ρ k ) =  µ + µ t
Dt
∂x j 
σk
 ∂k 

 + Gk − ρ ε
 ∂x j 
µ t  ∂ε 
D
∂ 
ε
ε2
(ρ ε ) =  µ +   + Ce1 Gk − ρ Cε 2
Dt
k
k
∂x j 
σ ε  ∂x j 
where Cµ = 0.09, Cε1 = 1.44, Cε 2 = 1.92,
‹
‹
‹
σ k = 1.0, σ ε = 1.3
The most widely-used engineering turbulence model for industrial applications
Robust and reasonably accurate; it has many sub-models for compressibility,
buoyancy, and combustion, etc.
Performs poorly for flows with strong separation, large streamline curvature,
and high pressure gradient.
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RANS Models – k–ω Models
µt = α∗ ρ
ρ
ρ
‹
‹
‹
‹
k
ω
∂u
Dk
∂
= τij i − ρ β∗ f β∗ k ω +
Dt
∂x j
∂x j
specific dissipation rate, ω

µt
 µ +
σk

ω ∂u
∂
Dω
= α τij i − ρ β f β ω2 +
∂x j
Dt
k ∂x j
 ∂k 


x
∂

j


µ t  ∂ω 


µ
+


σ ω  ∂x j 

ε 1
ω≈ ∝
k τ
Belongs to the general 2-equation EVM family. Fluent 6 supports the standard k–ω
model by Wilcox (1998), and Menter’s SST k–ω model (1994).
k–ω models have gained popularity mainly because:
z Can be integrated to the wall without using any damping functions
z Accurate and robust for a wide range of boundary layer flows with pressure
gradient
Most widely adopted in the aerospace and turbo-machinery communities.
Several sub-models/options of k–ω: compressibility effects, transitional flows and
shear-flow corrections.
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RANS Models – Reynolds Stress Model (RSM)
(
)
(
)
∂
∂
ρui′u ′j +
ρ u k ui′u ′j = Pij + Fij + DijT + Φ ij − ε ij
∂t
∂xk
Stress production
Rotation production
‹
‹
‹
‹
‹
Dissipation
Turbulent
diffusion
Pressure
Strain
Modeling required for these terms
Attempts to address the deficiencies of the EVM.
RSM is the most ‘physically sound’ model: anisotropy, history effects and
transport of Reynolds stresses are directly accounted for.
RSM requires substantially more modeling for the governing equations (the
pressure-strain is most critical and difficult one among them).
But RSM is more costly and difficult to converge than the 2-equation models.
Most suitable for complex 3-D flows with strong streamline curvature, swirl
and rotation.
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Standard Wall Functions
‹
Standard Wall Functions
z
Momentum boundary condition based on Launder-Spaulding law-of-thewall:
 y∗

U∗ = 1
∗
ln
E
y
 κ
(
z
z
z
)
for y∗ < yν∗
∗
∗
ν
for y > y
∗
where U =
U P Cµ1/ 4 kP1/ 2
U
2
τ
y∗ =
ρ Cµ1/ 4 kP1/ 2 y p
µ
Similar wall functions apply for energy and species.
Additional formulas account for k, ε, and ρ ui′u′j .
Less reliable when flow departs from conditions assumed in their
derivation.
„
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or body forces, low Re or highly 3D flows
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Standard Wall Functions
‹
Energy

ρ Cµ1 4 k1 2
Prt U P2 + (Pr− Prt )Uc2


2q
T∗ = 
14 12
2
ρ
Pr
C
k
U
1
1


µ
P
∗
∗
∗
Pr ln Ey + P + Pr y +
ln
E
y
t

  κ
κ
2q
[
]
( )
(
for y∗ < yt∗
)
for y∗ > yt∗
 Pr 3 4  

Pr 
P = 9.24   −1 1 + 0.28exp − 0.007 
Prt 
 
 Prt 

‹
Species

Sc y∗
for y∗ < yc∗

Y∗ =  1

∗
∗
∗
+
>
E
y
P
y
y
Sc
ln
for
t
c
c

  κ

(
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Non-Equilibrium Wall Functions
‹
Non-equilibrium wall functions
z
Standard wall functions are modified to account for stronger pressure
gradients and non-equilibrium flows.
„
„
‹
Useful for mildly separating, reattaching, or impinging flows.
Less reliable for high transpiration or body forces, low Re or highly 3D
flows.
The standard and non-equilibrium wall functions are options for all of
the k–ε models as well as the Reynolds stress model.
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Enhanced Wall Treatment
‹
Enhanced wall functions
z
z
z
Momentum boundary condition based on u + = e Γ u + + u + e1 Γ
lam
turb
a blended law-of-the-wall (Kader).
Similar blended wall functions apply for energy, species, and ω.
Kader’s form for blending allows for incorporation of additional physics.
„
„
‹
Pressure gradient effects
Thermal (including compressibility) effects
Two-layer zonal model
z
A blended two-layer model is used to determine near-wall ε field.
z
Domain is divided into viscosity-affected (near-wall) region and turbulent core
region.
ρy k
Œ Based on the wall-distance-based turbulent Reynolds number: Re y ≡
µ
Œ Zoning is dynamic and solution adaptive.
„ High Re turbulence model used in outer layer.
„ Simple turbulence model used in inner layer.
Solutions for ε and µT in each region are blended: λ ε (µ t )outer + (1 − λ ε )(µ t )inner
„
‹
The Enhanced Wall Treatment option is available for the k–ε and RSM models
(EWT is the sole treatment for Spalart Allmaras and k–ω models).
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Two-Layer Zonal Model
‹
The two regions are demarcated on a
cell-by-cell basis:
z
z
Turbulent core region
Re y > 200
Viscosity affected region
Re y < 200
z
z
y is the distance to the nearest wall
Zoning is dynamic and solution
adaptive
Re y ≡
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Turbulent Heat Transfer
‹
The Reynolds averaging of the energy equation produces an additional
term
ui′t ′
z
Analogous to the Reynolds stresses, this is the turbulent heat flux term. An
isotropic turbulent diffusivity is assumed:
ui′t ′ = − νT
z
Turbulent diffusivity is usually related to eddy viscosity via a turbulent
Prandtl number (modifiable by the users):
Prt =
‹
∂T
∂xi
νt
≈ 0.85 − 0.9
νT
Similar treatment is applicable to other turbulent scalar transport
equations.
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Menter’s SST k–ω Model Background
‹
Many people, including Menter (1994), have noted that:
z
z
z
The k–ω model has many good attributes and performs much better than
k–ε models for boundary layer flows.
Wilcox’ original k–ω model is overly sensitive to the free stream value of
ω, while the k–ε model is not prone to such problem.
Most two-equation models, including k–ε models, over predict turbulent
stresses in wake (velocity-defect) regions, which leads to poor
performance of the models for boundary layers under adverse pressure
gradient and separated flows.
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Menter’s SST k–ω Model Main Components
‹
The SST k–ω model consists of
z
z
Zonal (blended) k–ω / k–ε equations (to address item 1 and 2 in the
previous slide)
Clipping of turbulent viscosity so that turbulent stress stay within what is
dictated by the structural similarity constant. (Bradshaw, 1967) - addresses
item 3 in the previous slide
Outer layer
(wake and
outward)
Inner layer
(sub-layer,
log-layer)
k–ω model transformed from
standard k–ε model
3
k 2
ε=
l
Modified Wilcox’ε k–ω model
Wilcox’ original k-ω model
Wall
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Menter’s SST k–ω Model Blended equations
‹
The resulting blended equations are:

µT  ∂k 


 µ +

σ k  ∂x j 

∂ui
∂ 
µT  ∂ω 
Dω γ
1 ∂k ∂ω
2


(
)
ρ
= τij
− βρω +
µ
+
+
ρ
−
F
σ
2
1


1
ω2
∂x j 
σ ω  ∂x j 
Dt ν t ∂x j
ω ∂x j ∂x j
∂u
∂
Dk
ρ
= τij i − β* k ρ ω +
∂x j
∂x j
Dt
φ = F1 φ1 + (1 − F1 ) φ1 ; φ = β, σ k , σ ω , γ
Wall
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Large Eddy Simulation (LES)
N-S equation
∂ui ∂ui u j
1 ∂p
∂
=−
+
+
∂x j
ρ ∂xi ∂x j
∂t
 ∂ui
ν
 ∂x
j





ui (x, t ) = ui (x, t ) + ui′(x, t )
Instantaneous
component
Resolved
Scale
Subgrid
Scale
Filtered N-S
equation
‹
∂ui ∂ui u j
1 ∂p
∂
+
=−
+
∂t
∂x j
ρ ∂xi ∂x j
 ∂ui
ν
 ∂x
j

 ∂ τij
−
 ∂x
j

Filter, ∆
(
τij = ρ ui u j − ui u j
)
(Subgrid scale
Turbulent stress)
Spectrum of turbulent eddies in the Navier-Stokes equations is filtered:
z
z
z
The filter is a function of grid size
Eddies smaller than the grid size are removed and modeled by a subgrid
scale (SGS) model.
Larger eddies are directly solved numerically by the filtered transient N-S
equation
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LES in FLUENT
‹
LES has been most successful for high-end applications where the RANS
models fail to meet the needs. For example:
z
z
z
‹
Combustion
Mixing
External Aerodynamics (flows around bluff bodies)
Implementations in FLUENT:
z
Sub-grid scale (SGS) turbulent models:
„
„
„
„
z
‹
‹
‹
Smagorinsky-Lilly model
WALE model
Dynamic Smagorinsky-Lilly model
Dynamic kinetic energy transport model
Detached eddy simulation (DES) model
LES is applicable to all combustion models in FLUENT
Basic statistical tools are available: Time averaged and RMS values of solution
variables, built-in fast Fourier transform (FFT).
Before running LES, consult guidelines in the “Best Practices For LES”
(containing advice for meshing, subgrid model, numerics, BCs, and more)
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Detached Eddy Simulation (DES)
‹
Motivation
z
z
‹
For high-Re wall bounded flows, LES becomes prohibitively expensive to
resolve the near-wall region
Using RANS in near-wall regions would significantly mitigate the mesh
resolution requirement
RANS/LES hybrid model based on the Spalart-Allmaras turbulence
model:
2
D~
1  ∂
ν
ν
~~
~
= Cb1 S ν − Cw1 f w   +

Dt
 d  σ ~ν  ∂x j


∂~
ν 
~
(µ + ρ ν )
 + ...
∂x j 


d = min (d w , CDES∆ )
z
z
‹
One-equation SGS turbulence model
In equilibrium, it reduces to an algebraic model.
DES is a practical alternative to LES for high-Reynolds number flows in
external aerodynamic applications
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V2F Turbulence Model
‹
A model developed by Paul Durbin’s group at Stanford University.
z
z
z
‹
‹
‹
Durbin suggests that the wall-normal fluctuations v′2 are responsible for
the near-wall damping of the eddy viscosity
Requires two additional transport equations for v′2 and a relaxation
function f to be solved together with k and ε.
Eddy viscosity model is ν T ~ v′2 T instead of νT ~ k T
V2F shows promising results for many 3D, low Re, boundary layer
flows. For example, improved predictions for heat transfer in jet
impingement and separated flows, where k–ε models fail.
But V2F is still an eddy viscosity model and thus the same limitations
still apply.
V2F is an embedded add-on functionality in FLUENT which requires a
separate license from Cascade Technologies (www.turbulentflow.com)
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Stochastic Inlet Velocity Boundary Condition
‹
It is often important to specify realistic turbulent inflow velocity BC for accurate
prediction of the downstream flow:
ui (x, t ) = ui (x ) + ui′(x, t )
Instantaneous
‹
Coherent
+ Random
Different types of inlet boundary conditions for LES
z
z
z
No perturbations – Turbulent fluctuations are not present at the inlet.
Vortex method – Turbulence is mimicked by using the velocity field induced by many
quasi-random point-vortices on the inlet surface. The vortex method uses turbulence
quantities as input values (similar to those used for RANS-based models).
Spectral synthesizer
„
‹
Time
Averaged
Able to synthesize anisotropic, inhomogeneous turbulence from RANS results (k–ε, k–
ω, and RSM fields).
Can be used for RANS/LES zonal hybrid approach
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Initial Velocity Field for LES/DES
‹
‹
‹
Initial condition for velocity field does not affect statistically stationary
solutions
However, starting LES with a realistic turbulent velocity field can
substantially shorten the simulation time to get to statistically
stationary state
The spectral synthesizer can be used to superimpose turbulent velocity
on top of the mean velocity field
z
z
Uses steady-state RANS (k–ε, k–ω, RSM, etc.) solutions as inputs to the
spectral synthesizer
Accessible via a TUI command:
/solve/initialize/init-instantaneous-vel
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